Workplace layout method using convex polygon envelope

ABSTRACT

An improved method for laying out a workspace using the prior art crowding index, PDI, where the average interpoint distance between the personnel and/or equipment to be laid out, d act , can be determined. The improvement lies in using the convex hull area, A poly , of the distribution of points being laid out within the workplace space to calculate the actual crowding index for the workspace. The convex hull area is that area having a boundary line connecting pairs of points being laid out such that no line connecting any pair of points crosses the boundary line. The calculation of the convex hull area is illustrated using Pick&#39;s theorem with additional methods using the Surveyor&#39;s Area formula and Hero&#39;s formula also being described for calculating A poly . The improved crowding index is termed PDI poly  to distinguish it from the prior art crowding index, PDI act .

STATEMENT OF GOVERNMENT INTEREST

The invention described herein may be manufactured and used by or forthe Government of the United States of America for governmental purposeswithout the payment of any royalties thereon or therefore.

CROSS-REFERENCE TO RELATED APPLICATIONS

This patent application is co-pending with a related patent applicationentitled Approximation Method for Workplace Layout Using Convex PolygonEnvelope (Navy Case No. 77891) by the same inventor as this patentapplication.

BACKGROUND OF THE INVENTION

(1) Field of the Invention

This invention relates to a method for providing layouts of workplacesand more particularly to a generalized layout method for a workplacecontaining any number of spatial objects based on a crowding indexcalculated from a convex polygon envelope.

(2) Description of the Prior Art

In the inventor's previous patent entitled "Process Which Aids to theLaying Out of Locations of a Limited Number of 100, Personnel andEquipments in Functional Organization", U.S. Pat. No. 5,235,506, whichis incorporated into this disclosure in its entirety by reference, aprocess is described whereby the relationship among objects in aparticular space can be accurately determined to minimize crowding. Acrowding index, or Population Density Index (PDI), for the space, termedPDI_(act), is calculated and compared to theoretical minimum (PDI_(min))and maximum (PDI_(max)) values, such that PDI_(min) <PDI_(act)<PDI_(max). The formula for calculating PDI_(act) is as follows:##EQU1## where n=number of objects in the space;

A=the geometric area of the space; and

d_(act) =average Euclidean distance among all possible pairs of nobjects within the space.

The values of PD_(min) and PDI_(max) are given as follows: ##EQU2## and##EQU3## where Δ=the average Euclidean distance of all possible pairs ofpoints for a unit lattice, i.e., a lattice of n points uniformlydistributed in area A; and

c=an arbitrary constant which corresponds to the minimum possiblespacing between the objects, e.g., personnel standing shoulder toshoulder within a space would be spaced approximately one foot from headto head, so c would be equal to one foot.

It can be seen that PDI_(min) corresponds to a uniform distribution of npoints in the space, while PDI_(max) corresponds to a uniformdistribution of n points in the space with a minimum distance c betweeneach horizontal and vertical point.

For small workplace layouts, i.e., where the number of points do notexceed 100, a table of values for Δ is provided:

                  TABLE 1                                                         ______________________________________                                        EUCLIDEAN DISTANCE VALUES FOR                                                 SELECTED UNIT LATTICES (IN FT)                                                Lattice                Lattice                                                (n = Area)                                                                              Δ      (n = Area)                                                                             Δ                                       ______________________________________                                        2 × 1                                                                             1.00         7 × 4                                                                            2.97                                          2 × 2                                                                             1.14         7 × 5                                                                            3.19                                          3 × 1                                                                             1.00         7 × 6                                                                            3.43                                          3 × 2                                                                             1.42         7 × 7                                                                            3.68                                          3 × 3                                                                             1.63         8 × 2                                                                            2.97                                          4 × 2                                                                             1.71         8 × 3                                                                            3.09                                          4 × 3                                                                             1.90         8 × 8                                                                            4.20                                          4 × 4                                                                             2.14         9 × 2                                                                            3.29                                          5 × 2                                                                             2.01         9 × 3                                                                            3.41                                          5 × 3                                                                             2.19         9 × 9                                                                            4.72                                          5 × 4                                                                             2.41         10 × 2                                                                           3.62                                          5 × 5                                                                             2.65         10 × 3                                                                           3.72                                          6 × 2                                                                             2.32         10 × 4                                                                           3.88                                          6 × 3                                                                             2.48         10 × 5                                                                           4.07                                          6 × 4                                                                             2.69         10 × 6                                                                           4.27                                          6 × 5                                                                             2.92         10 × 7                                                                           4.50                                          6 × 6                                                                             3.18         10 × 8                                                                           4.74                                          7 × 2                                                                             2.65         10 × 9                                                                           4.98                                          7 × 3                                                                             2.78         10 × 10                                                                          5.24                                          ______________________________________                                    

In the inventor's related patent entitled "Two-Step Method ConstructingLarge-Area Facilities and Small-Area Intrafacilities EquipmentsOptimized by User Population Density", U.S. Pat. No. 5,402,335, thefollowing formula was provided for calculating Δ for any number ofpoints: ##EQU4## where C=the number of vertical points in each column ofthe unit lattice; and

R=the number of horizontal points in each row of the unit lattice, suchthat RC=the total number of points in the unit lattice.

The formula provides an exact solution for Δ, corresponding to thevalues given in Table 1.

As an example of using PDI to determine the crowding of a particularlayout, assume a workspace of 25 ft² (A=25) with a total of 12 objectsor personnel (n=12) which need to be laid out within the space. In usingTable 1 or equation (4), the row by column distribution of the latticepoints should be commensurate with the shape of the region A in whichthe lattice points reside. If the area is relatively square, i.e., 5×5,a corresponding distribution of 12 points would be 4×3, with Δ=1.90. Fora rectangular area of approximately 8.33×3, a corresponding distributionwould be 6×2, with Δ=2.32. Assuming a relatively square area and a 4×3distribution, the calculation of PDI_(min) from equation (2) yields(1/1.90)(12/25)≅0.25 and the calculation of PDI_(max) from equation (3)yields (1/c)(1/1.90)(12/25)^(1/2) ≅0.36, where c is taken as one foot asin the example of personnel standing shoulder to shoulder. Note that thevalue of PDI_(max) is seen to increase as c, or the minimum distancebetween points, becomes smaller, corresponding to the ability to packadditional points into the space. To determine PDI_(act), measurementsof the proposed distribution of points need to be taken and thosemeasurements used to calculate d_(act) as follows: ##EQU5## where d_(ij)=measured distance between point i and point j. Assuming a proposeddistribution where d_(act) =2.30, PDI_(act) from equation (1) yields(1/2.30)(12/25)^(1/2) ≅0.30. This would indicate the space is 20% morecrowded (0.25 vs. 0.30) than the theoretical minimum and 20% lesscrowded than the theoretical maximum (0.30 vs. 0.36).

It will be noticed that the total area, A, of the space is used incomputing PDI_(act), or the crowding index of the space. In actuality,the perceived crowding will depend on the area encompassed by the points(objects or personnel) within the space. As an example, assume fourpoints arranged in a square having sides approximately 4.4 units long.For such an arrangement, d_(act), as calculated in accordance withequation (5), is 2(d₁₂ +d₁₃ +d₁₄ +d₂₃ +d₂₄ +d₃₄)/(4(4-1))=5.0. This canbe compared to a rectangular arrangement having sides of 2.5 and 6 unitslong. Again d_(act) is calculated to be 5.0. For a 2×2 distribution, weobtain Δ=1.14 from Table 1. If we now assume an area of A=100 for bothdistributions and c=1, we can calculate PDI_(min), PDI_(max) andPDI_(act) from equations (2), (3) and (1), respectively.

    PDI.sub.min =(1/1.14)(4/100)=0.035

    PDI.sub.max =(1/1)(1/1.14)(4/100).sup.1/2 =0.175

    PDI.sub.act =(1/5)(4/80).sup.1/2 =0.040

Note that the various PDI's, or the crowding indices, have the samevalues for both the square and rectangular distributions of points.However, the perceived crowding of personnel separated by 2.5 units,which is approaching the minimum spacing of c=1, would probably begreater than those separated by 4.4 units. Carrying the example to itsextreme, a long, narrow rectangle can be formed where the two pointsforming the shorter side of the rectangle are at the minimum spacing"c". Again, the PDI values would remain the same, but the crowding atthe ends of the rectangle would probably be intolerable.

SUMMARY OF THE INVENTION

Accordingly, it is a general purpose and object of the present inventionto provide an improvement to the PDI method for laying out workplaceswhich better takes into account the distribution of points within anarea.

It is a further object of the present invention that the improvementbetter reflect the perceived crowding within the point distribution.

These objects are provided with the present invention by an improvedmethod of calculating the crowding index for a space, PDI_(act), whichaccounts for the distribution of points within the space. While the termd_(act) attempts to account for the spacing between points, it is to benoted in the example given above that the change from a squaredistribution of points to a rectangular distribution had no effect onthe crowding indices. However, with d_(act) held constant, there is achange in the area bounded by the points as the distribution moves froma square configuration (A_(square) =4.4×4.4=19.36) to a rectangular one(A_(rect) =2.5×6=15). The decrease in area is consistent with anincrease in the perceived crowding. It is proposed that a more accuratemeasure of the actual crowding index will utilize a measure of theactual polygonal space occupied by the distribution of points within thetotal area as well as the average Euclidean distance, d_(act) betweenthe points for which a layout is desired. This new measure can beexpressed as follows: ##EQU6## where A_(poly) the area of the polygonalspace occupied by the distribution.

The method of the present invention further provides for the calculationof A_(poly) based on the use of Pick's theorem, as further developedherein.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete understanding of the invention and many of the attendantadvantages thereto will be readily appreciated as the same becomesbetter understood by reference to the following detailed descriptionwhen considered in conjunction with the accompanying drawings whereincorresponding reference characters indicate corresponding partsthroughout the several views of the drawings and wherein:

FIG. 1 depicts an area A of 2×4 units with n=6 points distributedtherein;

FIG. 2 depicts a polygonal area containing the distributed points; and

FIG. 3 depicts a lattice overlaid on the polygonal area.

DESCRIPTION OF THE PREFERRED EMBODIMENT

Referring now to FIG. 1, there is shown a configuration of 6 points,denoted n₁ through n₆, arranged in a space of 8 square units. In usingthe improved PDI method, the values for PDI_(min) and PDI_(max) arecomputed in accordance with the prior art. Using Table 1 or equation(4), a 3×2 lattice is chosen, yielding a Δ of 1.42. Assuming a minimumdistance between objects of c=0.75, equations (2) and (3) yield valuesof 0.528 and 0.813 for PDI_(min) and PDI_(max), respectively. Theaverage Euclidean distance between points d_(act) is also determined inthe conventional manner, i.e., by measurements taken from time-lapseobservations. Using the distribution shown in FIG. 1, d_(act) ≅1.54. Todetermine PDI_(act) in the conventional manner, we use equation (1) withA=8. PDI_(act) is then determined to be ≅0.562. A PDI_(act) justslightly higher than PDI_(min) would indicate a non-crowded layout.Referring now to FIG. 2, the area to be used in calculating the improvedcrowding index, PDI_(poly), in accordance with the present invention isdepicted therein. The polygonal area A_(poly), is referred to as aconvex hull and is constructed as described in Computational Geometry:An Introduction, Preparata, F. P. and Shamos, M. I. (1985, pp. 104-106)New York: Springer-Verlag. Intuitively, the convex hull is constructedby imagining a rubber band stretched around all the points and, whenreleased, the band assumes the shape of the hull. If the points are thenconnected pairwise, no line falls outside the bounded figure. A numberof methods, well known in the art, are available for calculating theconvex hull area. In a previous paper, "Measuring The Areal Density Of AFinite Ensemble", Perceptual and Motor Skills, O'Brien, F. (1995, vol.81, pp. 195-200), the inventor discusses three such methods: (1) Pick'stheorem; (2) the Surveyor's Area formula; and (3) Hero's formula. Pick'stheorem will be used to illustrate the calculation of the area of theconvex hull, A_(poly), which will then be used in determining theimproved crowding index, PDI_(poly).

Referring now to FIG. 3, the convex hull is shown overlaid with a squarelattice of points such that each vertex of the hull meets a point of thelattice. It is anticipated the spacing between lattice points will bedetermined by overlaying the convex hull with lattices havingsuccessively smaller and smaller spacing. The process is continued untilthe lattice spacing is such that each vertex of the convex hull falls ona lattice point. Pick's theorem states: ##EQU7## where r=the spacingbetween lattice points;

i=the number of lattice points in the interior of the hull; and

b=the number of lattice points on the boundary.

FIG. 3 shows a lattice with r=0.25. Counting the number of latticepoints on the interior of the convex hull yields i=41 and the number oflattice points on the boundary gives b=5. A_(poly) from equation (7) is(0.25)² (41+5/2-1)=2.66. Using this value in equation (6) yields a valueof 0.98 for PDI_(poly). Since PDI_(poly) >PDI_(max), a crowded conditionis indicated. Looking to FIG. 3, we can see the points are tightlygrouped in the center of the rectangular area. The prior art crowdingindex, PDI_(act), accounted for the spacing of points within the groupthrough the term d_(act). However, the fact that the point distributionoccupies only a relatively small area (A_(poly) =2.66) within the totalarea (A=8.0) has no influence on the prior art crowding index. Asindicated in the above calculation, the use of the convex hull area termA_(poly) provides a new crowding index, PDI_(poly), which takes theactual area occupied by the distribution into account. The spacing ofpoints within the group is accounted for in the term d_(act) whencalculating PDI_(poly), in the same manner as in calculating PDI_(act).In the example given above, the new crowding index is found to be notonly greater than the prior art crowding index, but also greater thanPDI_(max), indicating the tight grouping of points does not make for themost efficient use of the total area A. Another way of looking at thisresult is to note that PDI_(poly) ' is independent of the area A beinglaid out. Changes in the total area A effect PDI_(min) and PDI_(max), orthe bounds of PDI_(poly) ', but the crowding index within the convexhull area is not effected by the changes to the total area A.

What has thus been described is an improvement to the prior art crowdingindex, or PDI, method for laying out workspace where the averageinterpoint distance between the personnel and/or equipment to be laidout, d_(act), can be determined. The improvement lies in using theconvex hull area, A_(poly), of the distribution of points being laid outwithin the space to calculate the actual crowding index for theworkspace. The convex hull area is that area having a boundary lineconnecting pairs of points being laid out such that no line connectingany pair of points crosses the boundary line. The calculation of theconvex hull area is illustrated using Pick's theorem. The improvedcrowding index is termed PDI_(poly) to distinguish it from the prior artcrowding index, PDI_(act). In the prior art, the distribution of pointswithin the workplace was taken into account in the crowding index,PDI_(act), solely through the average interpoint distance term d_(act).The use of the area bounded by the personnel or equipment being laid outin determining the improved crowding index, PDI_(poly), more fully takesinto account the distribution of points within the total area being laidout and also better reflects the perceived crowding within the pointdistribution.

While a preferred embodiment of the invention using Pick's theorem hasbeen disclosed in detail, it should be understood by those skilled inthe art that various other methods or formulae for calculating theconvex hull area, A_(poly), may be used. For example, the convex hullarea may be calculated using the Surveyor's Area formula or Hero'sformula.

When Cartesian coordinates are readily available for the points formingthe vertices of the convex hull, the Surveyor's Area formula may beused:

    A.sub.poly =1/2|(x.sub.1 y.sub.2 -x.sub.2 y.sub.1)+(x.sub.2 y.sub.3 -x.sub.3 y.sub.2)+ . . . +(x.sub.s y.sub.1 -x.sub.1 y.sub.s)|                                        (8)

where |·| indicates the absolute value and {(x₁ y₁), . . . (x_(s)y_(s))} are the Cartesian coordinates of the s boundary points of theconvex hull. When coordinate measurements are not easily available, butone is able to obtain the distances between the vertices of the convexhull, such as from an aerial photograph, then Hero's formula may beused. Hero's formula is based on summing the areas of non-overlappingtriangles within the convex hull. The area of each triangle iscalculated from: ##EQU8## where S₁, S₂ and S₃ are the lengths of thesides of triangle and C_(p) is the semiperimeter of the triangle, or (S₁+S₂ +S₃)/2. In general, s-2 triangles will result from a convex hullconsisting of s boundary points. The total area of the convex hull isthen given as: ##EQU9## As with the use of Pick's theorem to calculateA_(poly), the Surveyor's Area formula and Hero's formula are well knownin the art.

In light of the above, it is therefore understood that within the scopeof the appended claims, the invention may be practiced otherwise than asspecifically described.

What is claimed is:
 1. An improved method of determining a populationdensity index which aids in laying out the locations of any number ofobjects in a quadrilateral space where an exact average Euclideandistance between pairs of objects can be determined, the improved methodcomprising the steps of:distributing the objects in the space;determining a convex hull area for the distribution of the objects, theconvex hull area being bounded by a boundary line connecting pairs ofobjects such that no line connecting any pair of objects crosses theboundary line; determining the population density index for the spaceand the distribution of objects by the relationship ##EQU10## whered_(act) is the average Euclidean distance between pairs of objects, n isthe number of objects and A_(poly) is the convex hull area; andoptimizing the layout of the objects in the space based on thepopulation density index.
 2. An improved method of determining apopulation density index which aids in laying out the locations of anynumber of objects in a quadrilateral space where an exact averageEuclidean distance between pairs of objects can be determined, theimproved method comprising the steps of:distributing the objects in thespace; overlaying the space with a Cartesian coordinate system having anarbitrary origin; locating a first object having the minimum Xcoordinate and establishing the first object as a prime object; locatinga next object by extending a vertical line from the prime object androtating the vertical line in a clockwise manner about the prime object,the next object being that object first encountered by the verticalline; extending a boundary line section between the prime object and thenext object; establishing the next object as the prime object; locatinga next object by extending a sweep line from the prime object in thedirection of the boundary line section and rotating the sweep line in aclockwise manner about the prime object, the next object being thatobject first encountered by the sweep line; repeating the stepsbeginning with the step of extending a boundary line section until theboundary line is extended to the first point, forming a closed convexhull; determining the population density index for the space and thedistribution of objects by the relationship ##EQU11## where d_(act) isthe average Euclidean distance between pairs of objects, n is the numberof objects and A_(poly) is the convex hull area; and optimizing thelayout of the objects in the space based on the population densityindex.
 3. The improved method of claim 1 wherein the convex hulldetermining step further comprises the steps of:overlaying the convexhull area with progressively smaller square lattices until each vertexof the convex hull meets a point on the lattice; counting the number oflattice points in the interior of the convex hull; counting the numberof lattice points on the boundary line of the convex hull; anddetermining the convex hull area by the relationship ##EQU12## where ris the spacing between lattice points, i is the number of lattice pointsin the interior of the hull and b is the number of lattice points on theboundary line.
 4. The improved method of claim 2 wherein the convex hulldetermining step further comprises the steps of:overlaying the convexhull area with progressively smaller square lattices until each vertexof the convex hull meets a point on the lattice; counting the number oflattice points in the interior of the convex hull; counting the numberof lattice points on the boundary line of the convex hull; anddetermining the convex hull area by the relationship ##EQU13## where ris the spacing between lattice points, i is the number of lattice pointsin the interior of the hull and b is the number of lattice points on theboundary line.
 5. The improved method of claim 1 wherein the convex hulldetermining step further comprises the steps of:overlaying the convexhull with a Cartesian coordinate system; determining coordinates for theobjects lying on the boundary line of the convex hull; and determiningthe convex hull area from the relationship

    A.sub.poly =1/2|(x.sub.1 y.sub.2 -x.sub.2 y.sub.1)+(x.sub.2 y.sub.3 -x.sub.3 y.sub.2)+ . . . +(x.sub.s y.sub.1 -x.sub.1 y.sub.s)|

where |·| indicates the absolute value and {(x₁ y₁), . . . (x_(s)y_(s))} are the Cartesian coordinates of the objects lying on theboundary line, the number of objects lying on the boundary line beingdesignated s.
 6. The improved method of claim 2 wherein the convex hulldetermining step further comprises the steps of:overlaying the convexhull with a Cartesian coordinate system; determining coordinates for theobjects lying on the boundary line of the convex hull; and determiningthe convex hull area from the relationship

    A.sub.poly =1/2|(x.sub.1 y.sub.2 -x.sub.2 y.sub.1)+(x.sub.2 y.sub.3 -x.sub.3 y.sub.2)+ . . . +(x.sub.s y.sub.1 -x.sub.1 y.sub.s)|

where |·| indicates the absolute value and {(x₁ y₁), . . . (x_(s)y_(s))} are the Cartesian coordinates of the objects lying on theboundary line, the number of objects lying on the boundary line beingdesignated s.
 7. The improved method of claim 1 wherein the convex hulldetermining step further comprises the steps of:extending side linesconnecting pairs of objects lying on the boundary line to formnon-overlapping triangles within the convex hull; summing the area ofthe triangles to determine the total convex hull area.
 8. The improvedmethod of claim 7 wherein the area summing step further comprises thesteps of:measuring the lengths of the side lines of each triangle;determining a semiperimeter, C_(p), for each triangle equal to one halfof the sum of the lengths of the side lines for the triangle; anddetermining an area for each triangle from the relationship ##EQU14##where S₁, S₂ and S₃ are the lengths of the side lines of the triangle.9. The improved method of claim 2 wherein the convex hull determiningstep further comprises the steps of:extending side lines connectingpairs of objects lying on the boundary line to form non-overlappingtriangles within the convex hull; and summing the area of the trianglesto determine the total convex hull area.
 10. The improved method ofclaim 9 wherein the area summing step further comprises the stepsof:measuring the lengths of the side lines of each triangle; determininga semiperimeter, C_(p), for each triangle equal to one half of the sumof the lengths of the side lines for the triangle; and determining anarea for each triangle from the relationship ##EQU15## where S₁, S₂ andS₃ are the lengths of the side lines of the triangle.